From 60c10631fcd75a53304c6ccf4bd4f63b75d122d6 Mon Sep 17 00:00:00 2001 From: Robert Date: Wed, 24 Mar 2021 17:09:55 +0100 Subject: [PATCH] Finished determinant section --- chapters/linear_algebra.tex | 1 + chapters/sections/determinant.tex | 255 ++++++++++++++++++++++++++++++ script.pdf | Bin 464096 -> 0 bytes script.synctex(busy) | 0 script.tex | 7 +- 5 files changed, 262 insertions(+), 1 deletion(-) create mode 100644 chapters/sections/determinant.tex create mode 100644 script.synctex(busy) diff --git a/chapters/linear_algebra.tex b/chapters/linear_algebra.tex index 0615dd9..5858818 100644 --- a/chapters/linear_algebra.tex +++ b/chapters/linear_algebra.tex @@ -7,4 +7,5 @@ \subfile{sections/vector_spaces.tex} \subfile{sections/matrices.tex} + \subfile{sections/determinant.tex} \end{document} \ No newline at end of file diff --git a/chapters/sections/determinant.tex b/chapters/sections/determinant.tex new file mode 100644 index 0000000..202c099 --- /dev/null +++ b/chapters/sections/determinant.tex @@ -0,0 +1,255 @@ +\documentclass[../../script.tex]{subfiles} + +% !TEX root = ../../script.tex + +\begin{document} +\section{The Determinant} + +In this section we always define $A \in \field^{n \times n}$ and $z_1, \cdots, z_n$ the row vectors of $A$. We declare the mapping +\[ + \det: \field^{n \times n} \longrightarrow \field +\] +and define +\[ + \det(A) := \det(z_1, z_2, \dots, z_n) +\] + +\begin{defi} + There exists exactly one mapping $\det$ such that + \begin{enumerate}[(i)] + \item It is linear in the first row, i.e. + \[ + \det(z_1 + \lambda\tilde{z_1}, z_2, \cdots, z_n) = \det(z_1, z_2, \cdots, z_n) + \lambda \det(\tilde{z_1}, z_2, \cdots, z_n) + \] + + \item If $\tilde{A}$ is obtained from $A$ by swapping two rows + \[ + \det(A) = -\det(\tilde{A}) + \] + + \item $\det(I) = 1$ + \end{enumerate} + This mapping is called the determinant, and we write + \[ + \det A = \begin{vmatrix} + a_{11} & \cdots & a_{1n} \\ + \vdots & \ddots & \vdots \\ + a_{n1} & \cdots & a_{nn} \\ + \end{vmatrix} + \] +\end{defi} + +\begin{eg} + \[ + \begin{vmatrix} + a_{11} & a_{12} \\ + a_{21} & a_{22} + \end{vmatrix} + = a_{11}a_{22} - a_{21}a_{12} + \] + \begin{align*} + \begin{vmatrix} + a_{11} & a_{12} & a_{13} \\ + a_{21} & a_{22} & a_{23} \\ + a_{31} & a_{32} & a_{33} \\ + \end{vmatrix} + = &a_{11}a_{22}a_{33} + a_{12}a_{23}a_{31} + a_{13}a_{21}a_{32} \\ + &- a_{31}a_{22}a_{13} - a_{32}a_{23}a_{11} - a_{33}a_{21}a_{12} + \end{align*} +\end{eg} + +\begin{rem} + \begin{enumerate}[(i)] + \item Every determinant is linear in every row + \item If two rows are equal then $\det(A) = 0$ + \item If one row (w.l.o.g. $z_1$) is a linear combination of the others, so + \[ + z_1 = \alpha_2z_2 + \alpha_3z_3 + \cdots + \alpha_nz_n, ~~\alpha_1, \cdots, \alpha_n \in \field + \] + then + \begin{align*} + \det(z_1, z_2, \cdots, z_n) = &\alpha_2 \underbrace{\det(z_2, z_2, z_3, \cdots, z_n)}_0 + \\ + &\alpha_3 \underbrace{\det(z_3, z_2, z_3, \cdots, z_n)}_0 + \\ + &\cdots + \\ + &\alpha_n \underbrace{\det(z_n, z_2, z_3, \cdots, z_n)}_0 \\ + &= 0 + \end{align*} + + \item Adding a multiple of a row to another doesn't change the determinant + + \item Define + \begin{align*} + T_{ij} && \text{ swaps rows } i \text{ and } j \\ + M_i(\lambda) && \text{ multiplies row } i \text{ with } \lambda \ne 0 \\ + L_{ij}(\lambda) && \text{ adds } \lambda \text{-times row } j \text{ to row } i \\ + \end{align*} + Then + \begin{align*} + \det(T_{ij} A) &= -\det(A) \\ + \det(L_{ij}(\lambda) A) &= \det(A) \\ + \det(M_i(\lambda) A) &= \lambda\det(A) + \end{align*} + \end{enumerate} +\end{rem} + +\begin{lem} + Let $\det$ be the determinent, and $A, B \in \field^{n \times n}$. Let $A$ be in row echelon form, then + \[ + \det(AB) = a_{11} \cdot a_{22} \cdot \cdots \cdot a_{nn} \cdot \det(B) + \] +\end{lem} +\begin{proof} + First consider the case of $A$ not being invertible. This means that the last row of $A$ is a zero row, which in turn means that $\det(A) = 0$. + This also means that the last row of $AB$ is a zero row and therefore $\det(AB) = 0$. + + Now let $A$ be invertible. This means that all the diagonal entries are non-zero. It is possible to bring $A$ into diagonal form without changing + the diagonal entries themselves. So, w.l.o.g. let $A$ be in diagonal form: + \begin{equation} + A = M_n(a_{nn}) \cdot \cdots \cdot M_2(a_{22})M_1(a_{11}) I + \end{equation} + and thus + \begin{equation} + \begin{split} + \det(AB) &= \det(M_n(a_{nn}) \cdot \cdots \cdot M_2(a_{22})M_1(a_{11}) B) \\ + &= a_{nn} \cdot \cdots \cdot a_{22} \cdot a_{11} \det(B) + \end{split} + \end{equation} +\end{proof} + +\begin{rem} + For $B = I$ this results in + \[ + \det(A) = a_{11} a_{22} \cdots a_{nn} + \] +\end{rem} + +\begin{thm} + Let $A, B \in \field^{n \times n}$. Then + \[ + \det AB = \det A \cdot \det B + \] +\end{thm} +\begin{proof} + Let $i, j \in \set{1, \cdots, n}$ and $\lambda \ne 0$. Then + \begin{subequations} + \begin{equation} + \det(T_{ij} AB) = -\det(AB) + \end{equation} + \begin{equation} + \det(L_{ij}(\lambda) AB) = \det(AB) + \end{equation} + \end{subequations} + Bring $A$ with $T_{ij}$ and $L_{ij}(\lambda)$ operations into row echelon form. Then + \begin{equation} + \det(AB) = a_{11}a_{22} \cdots a_{nn} \cdot \det(B) + \end{equation} + and therefore + \begin{equation} + \det(AB) = \det A \cdot \det B + \end{equation} +\end{proof} + +\begin{cor} + \[ + A \in \field^{n \times n} \text{ invertible } \iff \det A \ne 0 + \] +\end{cor} +\begin{proof} + Row operations don't effect the invertibility or the determinant (except for the sign) of a matrix. Therefore we can limit ourselves to matrices in + row echelon form (w.l.o.g.). Let $A$ be in row echelon form, then + \begin{equation} + \begin{split} + \det A \ne 0 &\iff a_{11} a_{22} \cdots a_{nn} \ne 0 \\ + &\iff a_{11} \ne 0, a_{22} \ne 0, \cdots, a_{nn} \ne 0 \\ + &\iff A \text{ invertible since diagonal entries are non-zero} + \end{split} + \end{equation} +\end{proof} + +\begin{thm} + \[ + \det A = \det A^T + \] +\end{thm} +\begin{proof} + First consider the explicit representation of row operations: + \begin{subequations} + \begin{equation} + T_{ij} = \kbordermatrix{ + & & j & & i & \\ + & 1 & & & & \\ + i & & 0 & & 1 & \\ + & & & 1 & & \\ + j & & 1 & & 0 & \\ + & & & & & 1 \\ + } + \end{equation} + \begin{equation} + L_{ij}(\lambda) = \kbordermatrix{ + & & & & j & \\ + & 1 & & & & \\ + i & & 1 & & \lambda & \\ + & & & 1 & & \\ + & & & & 1 & \\ + & & & & & 1 \\ + } + \end{equation} + \end{subequations} + Thus we can see + \begin{subequations} + \begin{equation} + \det(T_{ij}) = \det(T_{ij}^T) = -1 + \end{equation} + \begin{equation} + \det(L_{ij}(\lambda)) = \det(L_{ij}(\lambda)^T) = 1 + \end{equation} + \end{subequations} + Let $T$ be one of those matrices. Then + \begin{equation} + \begin{split} + \det((TA)^T) &= \det(A^T \cdot T^T) \\ + &= \det A^T \cdot \det T^T \\ + &= \det A^T \cdot \det T \\ + \end{split} + \end{equation} + and + \begin{equation} + \det TA = \det A \cdot \det T + \end{equation} + And therefore + \begin{equation} + \det((TA)^T) = \det(TA) \iff \det A^T = \det A + \end{equation} + Now w.l.o.g. let $A$ be in row echelon form. Let $A$ be non-invertible, i.e. the last row is a zero row. Thus $\det A = 0$. + This implies that $A^T$ has a zero column. Row operations that bring $A^T$ into row echelon form (w.l.o.g.) perserve this zero column. Therefore the + resulting matrix must also have a zero column, and thus $\det(A^T) = 0$. + + Now assume $A$ is invertible, and use row operations to bring $A$ into a diagonalised form (w.l.o.g.). For diagonalised matrices we know that + \begin{equation} + A = A^T \implies \det A = \det A^T + \end{equation} +\end{proof} + +\begin{rem} + Let $A_{ij}$ be the matrix you get by removing the $i$-th row and the $j$-th column from $A$. + \[ + \det A = \sum_{i=1}^n (-1)^{i+j} \cdot a_{ij} \cdot \det(A_{ij}), ~~j \in \set{1, \cdots, n} + \] +\end{rem} + +\begin{rem}[Leibniz formula] + Let $n \in \natn$, and let there be a bijective mapping + \[ + \sigma: \set{1, \cdots, n} \longrightarrow \set{1, \cdots, n} + \] + $\sigma$ is a permutation. The set of all permutations is labeled $S_n$, and it contains $n!$ elements. Then + \[ + \det A = \sum_{\sigma \in S_n} \sgn(\sigma) \prod_{i=1}^n a_{i, \sigma(i)} + \] + A permutation that swaps exactly two elements is called elementary permutation. Every permutation can be written as a number of consecutively executed elementary permutations. + \[ + \sgn(\sigma) = (-1)^k + \] + where $\sigma$ is the permutation in question and $k$ is the number of elementary permutations it consists of. +\end{rem} +\end{document} \ No newline at end of file diff --git a/script.pdf b/script.pdf index 14020df6bc8a68e336fa6f5674fff8318a00e5a6..e69de29bb2d1d6434b8b29ae775ad8c2e48c5391 100644 GIT binary patch literal 0 HcmV?d00001 literal 464096 zcmce;1z40((?4t=-5{aF(jhKv081zm(hY)icZYOIgGefkgoJ{Wv>*-AAtK!^ErK8* z-?A1eKKj1z^ZmIl?rV4Nd+wPtXU;jlnKNhTWQByUf`IJkbUkmH2GOAaHh`6`DLOAN zfK>s&Dq^K*1z;5dFzOlWSlb)e0zpiG>(|i@EcK5c;{5tBqdJ6>1OCsc&c?zV3n4P1WB+LS2uToPuho^YWP2l@`nTnMhMXm2KR;Uv3%cTM~H}c05R&n{frny2%!;r z_#yrep&@@8Nnqel<_S4LO2|MMM#NIGhzgv)`o9#A}9?{Ht2^I5N|+8 z{-xR9?FI6uf%VPQf0p~yVf8~nKRW)0p?(vO|5!4FP5o#IVgw>&)A=qD^iPBDn*~FD zwqWR~!%ykE@_x)N2n8XyeUk~HAnW5t5%Qj}iyz|rZ9qYOwnylx14`;U4}{_W=#CRC zk01Fa*dI($M^MRXM6`5y!Ei{<@j_#XzM8uZu* zV0CiWU9R#WzG1Cf^(omJAn(#YNz0A}L=eECC%KW+|Rf#GE4I`#%a26|Td2I%m$ z*v{V8K*s{zY1l8E->iWR({Fw6A~$_cOSCzIM3~S(?+Aj-Kgn(T5HaY@}W85uU-gQXn$8Kxwg9S$Lzkm30qTZ43XqkJ zAFt`r5ZN3bAA7x{=g8$ia3unFTJr2I#DUR+jb#miBf4g#P5wS>+Av ztnS$A8Nh!z9trSYqzv>;bOfxN0BXlSaKM*sPA;$pI{XE;a7qCC@er0j*5y}T>_28B zbU66=-eI>@Q~@9`6v)X118}mj0ogcV0Lwe(<{Bqyzm2BjUi^)g9YM>%0iT0GoWIll z7`f;Gc*D%CY!$3^^b7#U?}QtStd2e00xSW?;=#Y(bg~ywu!kE1;Mj8H0dNz6>jH2L z-Gsqh0QTd7cWh=5HURtad?|lyY^<`jR(c8s_5d~b`-Fr6tcnIs_JETu;5;G4H#=m9 zg8tbK`=hn264(fvCbpOkj47;Fmpz@VM&2kUJvUYD)y%$D-grM8^_e#P_DFXJ+I>LM zaPI5Z#F9v(f*uQx%ov9O!9XJWGLIHwBMBW`)HFz^tVaNeXquFNx}lKx{rBuI>L?Sq zr!ZZKV}N^)JTKi#yh?X71YfXo9i1>k99LS(Et>A>*`Sypx$^u02Oh$+D;E89Ox;MO z+b*!F-R#Y`G5qfexL=6Hy$;ri4HR%-Y$Kc!PkCTN-J}_ZWFdTZBYWsUQ!xvU)y+sn zc^B|Kw)y^GDJtHFsXOf>pLVgiZiT1mOOyAuPYE~wTv z$`0$^jC#B6Yz*ehGgvW2%4`CVj*fc^=R)kJs%#rz=tM3f*Bg~;o{hO%<-CruoHxy! zYOIO}r`8b}KI-qV4&T}z@ApgacfVyRG119x>=56uNrIWu9p}u3>^k^PfbT%^Mu55% zG7Fxt9r1wK0L2of-t0)rYOU|R8XRC)oyNnGqE+CF{^qe#2j&YY0_wglHrMxJyRF0+ z*U=}6d3N~BK}M`0*GujnB(KB)T$yX!I@XPrM{Fr3Fh(3TOqO@sZ&7t__u{Z*?0N1N zy$?!?VG{RR?s1dHi&{E-H9OV0us2Oilg_U5`>^Ktwjdr4ms5u|!W@4Z)*QzZ>pu@` zR(?xMEBN%F24Fv4S>yri2yy`XmsywtF-O2B1;jm$FGGvt%YftfGJ-h13?2@|Gy&lF z@|Ke$2Jzyb=OzSGbXJ8sy7q{ohsHhwDHZ)Iw9Eaeg1zIjFSrpVut|0>@c{R z*a1)&49E!v{Y9`ZIQWNP2wV6s_2V~k9(!P%g;nVtuo)ZywVL z3;X7!{1^1~b*g|sVL&c8&Vs-N19PzhAn>`2{Xb7(-(063a&exVDd6xEvA)7V;1}rn zpW!EBUHv-C{2$>be1LF%MXADY6wdh-zzV~I6P#ZbRk+Ci07Xw~`KwFB1%v)G6wOeU zvK!&ReYvl^fhsM5G;t=ugva8NQD)yfe&;N18M&(e)h=0makPWXUZf~cIknf;45v19EeswwtA#z@dtl$U(sixNu5HvGy&eF!Ep zl9?S3Juh=aAz{>K^oG@a=P!ov)nt=DTc+g?=kQocGZy{87UY9bp-Ph!4;Q zD2%xri&^YzI+sE&qXZ3d5Sp$j)heT}Gu`5$%pG=nHH05+h$BQEUW71&I+O!g)LLhxP(uMmDt9<)g`IJ$xo#h&*$+4;x;r|XS~PPSiHsh!Nn<5+@HO; 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